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The metric construction of a compatible almost complex structure is continuous in the auxiliary Riemannian metric, and recovers $J$ when applied to $g_J=\omega (\cdot ,J\cdot )$. Interpolating $g_J$ linearly to a fixed auxiliary metric therefore contracts the space of compatible almost complex structures. In particular it is nonempty and path connected.

A compatible almost complex structure is a smooth endomorphism $J:TX\to TX$ satisfying $J^2=-I$, preserving $\omega$, and making $g_J(u,v)=\omega (u,Jv)$ a positive-definite inner product. Preservation and $J^2=-I$ make this bilinear form symmetric.

To construct one, choose any Riemannian metric $h$ and define $A$ by $h(Au,v)=\omega (u,v)$. The endomorphism $A$ is skew-adjoint and invertible. Consequently $-A^2$ is positive definite and self-adjoint. Let $B=(-A^2{)}^{1/2}$ be its positive square root and setThe spectral theorem gives $B$ fibrewise, and the positive-square-root operation is smooth on positive-definite matrices, so this defines a smooth endomorphism. Since $A$ commutes with $B$, $J^2=-I$. Also $J^{\ast }=-J$, so $J^{\ast }J=I$. Commutation then givesThis is the metric construction of a compatible almost complex structure, proving nonemptiness.

If we start with $h=g_J$ for an already compatible $J$, then $h(Ju,v)=\omega (u,v)$, so $A=J$, $B=I$, and the construction recovers $J$. For two compatible structures, linearly interpolate their metrics,and apply the construction to $h_t$. This is a continuous smooth-in-$t$ path with endpoints $J_0,J_1$. Thus the space is nonempty and path connected. Interpolating instead to a single fixed auxiliary metric actually gives the contractibility of compatible almost complex structures.

An almost complex structure $J$ is compatible with $\omega$ if $J^2=-I$, $\omega (Ju,Jv)=\omega (u,v)$, and $g_J(u,v)=\omega (u,Jv)$ is a positive-definite inner product. To construct one, choose a Riemannian metric $h$ and define $A$ by $h(Au,v)=\omega (u,v)$. The map $A$ is invertible and skew-adjoint, so $-A^2$ is positive definite. Its positive square root $P=(-A^2{)}^{1/2}$ commutes with $A$, andThe same identities show that $J$ preserves $\omega$. The positive square root depends smoothly on $A$, giving a smooth compatible almost complex structure.

For a compatible $J$, applying this metric construction of a compatible almost complex structure to $g_J$ returns $J$. Given $J_0,J_1$, apply it to $(1-t)g_{J_0}+t{g}_{J_1}$ to obtain a path joining them. In fact interpolation to any fixed auxiliary metric gives a contraction. Thus the space is nonempty and connected, and even contractible, by contractibility of compatible almost complex structures.

Regularity of a J-holomorphic curve $u$ means that its linearized Cauchy–Riemann operator $D_u$ is surjective. Differentiating in coordinates givesup to the harmless factor $1/2$ in the convention for ${\bar{\partial }}_J$. For constant $u$, the last term vanishes, $J(u)$ is constant, and the pulled-back tangent bundle is ${\mathbb{CP}}^1\times T_{u}M$. Identifying $T_{u}M$ with ${\mathbb{C}}^n$, this is a direct sum of $n$ copies of the Dolbeault operator on the trivial line bundle. Its cokernel is $H^{0,1}({\mathbb{CP}}^1,\mathcal{O})=0$, by Dolbeault cohomology of the projective line. Hence constant maps are regular for every compatible $J$.

The Energy identity for a J-holomorphic curve says that the energy is ${\int }_{{\mathbb{CP}}^1}{u}^{\ast }\omega$. If the homology class is zero, this integral is zero, forcing $du=0$. Thus every zero-class sphere is constant and regular.

In real dimension four, the Fredholm index of the parametrized sphere operator is $4+2c_1(A)$. At a regular nonconstant sphere, quotienting by the six-dimensional Möbius transformation group gives a local moduli space, or an orbifold for finite stabilizers, of real dimensionThis is the dimension formula for regular J-holomorphic spheres.

For a simple J-holomorphic sphere in class $A$, the adjunction inequality for a simple J-holomorphic sphere gives $c_1(A)\le A^2+2$. If $A^2\le -2$, the displayed moduli dimension is negative, so such a regular simple J-holomorphic sphere cannot exist. To cover nonsimple maps as well, interpret regularity here as regularity for all nonconstant maps under discussion. A degree-$m$ cover, $m\ge 2$, of a simple J-holomorphic sphere in class $B$ has $A=mB$, with $B^2\le -1$ and $c_1(B)\le 1$. Its local family of rational covering maps of the projective line modulo domain reparametrization has real dimension $4m-4$, while the predicted dimension isIt therefore cannot be regular. This proves there are no regular spheres in a class with square at most minus two, by negative-square exclusion under full sphere regularity. Regularity only for simple curves would not suffice: multiple covers of an exceptional minus-one sphere are the counterexample.

An almost complex structure $J$ on $(M,{\omega }_M)$ is a compatible almost complex structure when $J^2=-I$, ${\omega }_M(Ju,Jv)={\omega }_M(u,v)$, andis a positive-definite inner product. To prove existence, choose any Riemannian metric $h$ and define $A$ by ${\omega }_M(u,v)=h(Au,v)$. The metric construction of a compatible almost complex structureis smooth and compatible, so the space is nonempty.

Identify the compact symplectic manifold $N$ with its image under the symplectic embedding. Along $N$ there is a symplectic splittingChoose compatible almost complex structures on both summands and take their direct sum. Its associated metric makes the two summands orthogonal. Extend this metric from the closed submanifold $N$ to all of $M$ using a partition of unity, and apply the metric construction again. Along $N$ it recovers the prescribed direct sum, so the resulting global compatible $J$ satisfies $J(TN)=TN$. This is the relative extension of a compatible almost complex structure.

For the two compact complex curves, use the supplied holomorphic coordinates at their transverse intersection. There $C_1\cup C_2$ is $\{xy=0\}$. Replace it in a small ball by the complex annulus $\{xy=ϵ\}$ and use a cutoff in a surrounding annulus to rejoin the unchanged curves. For sufficiently small nonzero $ϵ$, the result is an embedded symplectic surface $\Sigma$. This local replacement is a bordism between the old and new cycles, soThis is the symplectic smoothing of a positive transverse node.

If a compatible almost complex structure is prescribed along a closed symplectic submanifold $N\subset M$ and preserves $TN$, it extends to one on $M$. Choose a compatible structure separately on $TN$ and its symplectic normal bundle, use the associated metric along $N$, extend that metric to $M$, and apply the metric construction of a compatible almost complex structure.